mitt ulrich fotheringham, schott ag: relaxation processes...
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MITT Ulrich Fotheringham, SCHOTT AG: Relaxation Processes in Glass and Polymers, Lecture 5
Special Topics in Relaxation in Glass and Polymers
Lecture 5: Viscoelasticity IShear
Dr. Ulrich Fotheringham
Research and Technology Development
SCHOTT AG
MITT Ulrich Fotheringham, SCHOTT AG: Relaxation Processes in Glass and Polymers, Lecture 5
2.1 Maxwell Modell
Mechanical model consisting of a spring and a dash-pot in a series.
Continuous strain:
Constant elongation:
derived from:
The time constant τ=(A×η)/(D×d) is called relaxation time.
z
x y
F
z
t
dA
FDF
z zz ⋅⋅
+=η
xDFx ⋅=dtdy
dAFy ⋅⋅= η
zyx =+
zyx FFF ==
0=+dtdy
dtdx
01=⋅+⋅
ηzz F
Ad
dtdF
D
)A/()dD(tz econst)t(F ⋅⋅⋅−⋅= η
Basic viscoelastic models I: Maxwell Model
Mechanical model consisting of a spring and a dash-pot in a series.
Viscoelastic behaviouraccording to the Maxwell-model means that the material considered responds like an elastic body on short time scales and like a viscous fluid on long time scales.
Continuous strain:
Constant elongation:
derived from:
The time constant τ=(A×η)/(D×d) is called relaxation time (force response to elongation).
MITT Ulrich Fotheringham, SCHOTT AG: Relaxation Processes in Glass and Polymers, Lecture 5
2.1 Maxwell Modell
Mechanical model consisting of a spring and a dash-pot in a series.
Continuous strain:
Constant elongation:
derived from:
The time constant τ=(A×η)/(D×d) is called relaxation time.
Basic viscoelastic models II: Kelvin-Voigt Model
Mechanical model consisting of a spring and a dash-pot in parallel.
Viscoelastic behaviour according to the Kelvin-Voigt-model means that the material responds like a rigid body on short time-scales and like an elastic body on long time-scales.
Both the Maxwell and the Kelvin-Voigt model are necessary to describe glass.
The case of an instantaneous, fixed elongation may not be realised with a Voigt-model.
Continuous strain:
derived from:
The time constant τ=(A×η)/(D×d) is called retardation time (elongation response to force).
xDFx ⋅=
dtdy
dAFy ⋅⋅= η
zx
y
F
z
zyx ==
yxz FFF +=
zFdtdz
dAzD =⋅⋅+⋅ η
( ))/()(1 η⋅⋅⋅−−⋅= ADdtz eDF
z
MITT Ulrich Fotheringham, SCHOTT AG: Relaxation Processes in Glass and Polymers, Lecture 5
Some exercises:
1. Consider “Basic viscoelastic models II: Kelvin-Voigt Model”. A Kelvin-Voigt Model acting together with the tyre as additional spring is what you find as damper in cars.
Introduce numbers:
Spring:
Calculate τ of the Kelvin-Voigt Model.
2. Consider again “Basic viscoelastic models I: Maxwell Model”. At the bottom, it is said that the formula
giving Fz(T) for constant elongation follows from the homogeneous linear differential equation
Find the solution via Laplace-Transform. Write down the subsidiary equation.
01=⋅+⋅
ηzz F
Ad
dtdF
D
)A/()dD(tz econst)t(F ⋅⋅⋅−⋅= η
mm/N20D =
m/sN2000dA
⋅=⋅η
MITT Ulrich Fotheringham, SCHOTT AG: Relaxation Processes in Glass and Polymers, Lecture 5
Shear Viscoelasticity and Bulk Viscoelasticity
Combining Maxwell- and Kelvin-Voigt-elements, one may describe the viscoelasticbehaviour of inorganic glasses and polymers. However, one has to distinguish between shear on one side and dilatation/compression on the other.
Shear
Mostly not the angle ε’ describing the change of γ’ is referred to but the angle εdescribing the change of γ. One has to introduce a factor of 2 for compensation.
Next step: drop the distinction between “x” and “y”. It is all in one.
A
x,y,z
d
F
ε
' γ
'
γ
ε
xxx GxDF 'εσ ⋅=→⋅=
dtd
dtdy
dAF y
yy
'εηση ⋅=→⋅⋅=
xxx 2GxDF εσ ⋅⋅=→⋅=dt
ddtdy
dAF y
yy
εηση ⋅⋅=→⋅⋅= 2
MITT Ulrich Fotheringham, SCHOTT AG: Relaxation Processes in Glass and Polymers, Lecture 5
Relations for Maxwell- and Kelvin-Voigt-model in case of shear
For the simple Maxwell-model and constant shear stress this leads to:
J(t) is called compliance (describes strain response to stress).
For the simple Maxwell-model and constant shear strain this leads to:
G(t) is the thus defined time-dependent shear modulus (stress response to strain).
For the simple Kelvin-Voigt-model and constant shear stress this leads to:
The combination of simple Kelvin-Voigt-model and constant shear strain does not exist.
0000 )(:
221
22σσ
ηησσ
ε ⋅=⋅
+=⋅+= tJt
Gt
G
00)/()/(
0 )(2:)2(2)( εεεσ ηη ⋅⋅=⋅⋅⋅=⋅⋅⋅= ⋅−⋅− tGeGeGt GtGt
( ) ( ) 00)/()/(0 )(:1
211
2σσ
σε ηη ⋅=⋅
−⋅=−⋅= ⋅−⋅− tJe
Ge
GGtGt
MITT Ulrich Fotheringham, SCHOTT AG: Relaxation Processes in Glass and Polymers, Lecture 5
Visco-Elastic Characterization of Glass
Pyrex® and BK7
Hemanth Chaitanya KadaliMechanical [email protected]
AdvisorDr. Vincent Blouin
Materials Science and Engineering
Committee Members
Dr. Paul Joseph Dr. Lonny Thompson
Mechanical Engineering
Clemson University, SC-29634
1
Shear relaxation experiments from Clemson University I
Kadali et al. have built a spring relaxometer as introduced by S. Rekhson et al.
MITT Ulrich Fotheringham, SCHOTT AG: Relaxation Processes in Glass and Polymers, Lecture 5
II Creep Recovery Setup
6
Shear relaxation experiments from Clemson University II
To impose pure shear on a sample, torsion is an appropriate way. The elongation of a helical spring implies torsion of every cross-section => spring elongation means shear.
III Creep Testing on Helical Spring sample
7
MITT Ulrich Fotheringham, SCHOTT AG: Relaxation Processes in Glass and Polymers, Lecture 5
II Creep testing
8
An Example of an LVDT Result
9
Shear relaxation experiments from Clemson University III
MITT Ulrich Fotheringham, SCHOTT AG: Relaxation Processes in Glass and Polymers, Lecture 5
1st Realistic Picture of Shear Relaxation: Burger-Model
As we have seen, the behaviour of inorganic glasses is more complicated and cannot be described with either the Maxwell- or the Voigt-model alone.
The simplest representation is by a Burger-model:
G 2η2
G 1
η1
t
εcreep
delayed elasticity
Instantaneous elasticity
MITT Ulrich Fotheringham, SCHOTT AG: Relaxation Processes in Glass and Polymers, Lecture 5
Burger-Model (continued)
In case of constant stress one gets:
The case of constant strain is more complicated and requires a careful derivation:
λ+, λ- are the roots of the equation G1×G2+(G2×η2+G1×η2+G2×η1)×λ+η1×η2×λ2=0.
( ) 00)/(
122
)(:12
122
111 σσ
ηε η ⋅=⋅
−⋅+
+= ⋅ tJe
Gt
G BurgerGt
⋅
+−⋅
+⋅
−⋅⋅
= ⋅−
⋅+
−+
−+ t
1
1t
1
102 eGeG)(
G2 λλη
λη
λλλε
σ
21
212112212222122122
2)()22()(
ηηηηηηηηηηη
λGGGGGGGGG −+++±++−
=±
MITT Ulrich Fotheringham, SCHOTT AG: Relaxation Processes in Glass and Polymers, Lecture 5
Kohlrausch-Kinetics for Shear Relaxation
An even better coincidence of theoretical and experimental data than with the Burger model is obtained, if the single exponential function from above is replaced with a stretched exponential or Kohlrausch(-Williams-Watts)-function.
For constant stress one gets:
J(t) is the time-dependent compliance; τd and bd are retardation parameters.
For constant strain one gets
G(t) is the time-dependent shear modulus; τx and bx are relaxation parameters.
1 2 3
σ/σ(0)1
0,5Kohlrausch
single exponential function
t /τ
( )00
/t
00)t(J:e1
G21
G21
2t
G21 db
d σση
ε τ ⋅=⋅
−⋅
−++= −
∞
( )0
/t00 )t(G:eG2)t(
xbx εεσ τ ⋅=⋅⋅⋅= −
( ) 1b0,eb/t <<− τ
MITT Ulrich Fotheringham, SCHOTT AG: Relaxation Processes in Glass and Polymers, Lecture 5
Boltzmann´s superposition principle, relaxation retardationIn general (not only for shear) it is assumed that the stress effects arising from strain contributions imposed at different times overlay without interfering and vice versa:
One may write introducing relaxation function Ψ
and retardation function Φ
Laplace-Transform yields:
and
This allows the mutual conversion of relaxation and retardation parameters.
( )( )( ) ( )( )
tdtd
)t(deG2eG2)t(t
00
/ttttimetheat0
xbx/ttxbx ′⋅
′′
⋅⋅⋅≈⋅⋅⋅= ∫∑′−−′−−
′ε
ε∆σττ
tdtd
)t(dttG2)t(t
0 x0 ′⋅
′′
⋅
′−⋅⋅= ∫
ετ
Ψσ
tdtd
)t(dtt1G21
G21
2tt
G21)t(
t
0 d00′⋅
′′
⋅
′−−⋅
−+
′−+= ∫
∞
στ
Φη
ε
( ) ( ) ( )εΨσ LsLG2L 0 ⋅⋅⋅⋅= ( ) ( ) ( )σΦη
ε LLsG21
G21
s21
G21L
0⋅
⋅⋅
−−+=
∞∞
MITT Ulrich Fotheringham, SCHOTT AG: Relaxation Processes in Glass and Polymers, Lecture 5
Relaxation and Viscosity
Consider the special case of constant stress in the long-time limit. According to the nature of the Laplace-Transform
only the L-values belonging to small s-values are sensitive to things happening on a broad time scale. This means for the retardation formula:
Replacing the low s, high t – limit of L(σ)/L(ε) with the expression that follows from the relaxation formula gives:
( ) ( ) ´dtet́ffL ´stt
0⋅⋅= −∫
( ) ( ) ( ) ( ) ( )ση
εσΦη
ε Llimk2
1LlimLLsG21
G21
s21
G21L
t,0st,0s0 ∞→→∞→→∞∞⋅=⇒⋅
⋅⋅
−−+=
( )
( ) ( ) ( ) ´dtt́´dtet́limLlimG
s2kLlimG2
0
´stt
0t,0st,0s0
t,0s0
⋅=⋅⋅==⇒
⋅⋅=⋅⋅⋅
∫∫∞
−
∞→→∞→→
∞→→
ΨΨΨη
ηΨ
MITT Ulrich Fotheringham, SCHOTT AG: Relaxation Processes in Glass and Polymers, Lecture 5
Addenda
Prony Series
For computational purposes, the Kohlrausch-function may be represented by a number of single exponentials (Prony-series):
In order to allow for the comparatively fast relaxation for t < τ and the comparatively slow relaxation for t > τ, the Prony-series must comprehend both τi < τ and τi > τ.
Temperature dependence of the relaxation times
As known, the temperature dependence of shear viscosity is well described by the Arrhenius law for a limited temperature range such as the one of glass transition:
Consequently, it is also used for shear relaxation and retardation times:
TRH
e ⋅= 0ηη
TRH
0e ⋅=ττ
( ) 1v,i0v,evei
iii
/ti
/t ib
=∀>⋅= ∑∑ −− ττ
MITT Ulrich Fotheringham, SCHOTT AG: Relaxation Processes in Glass and Polymers, Lecture 5
Some exercises:
1. Consider the correspondence of the Maxwell model with spring and dashpot and the behaviourof a viscoelastic material. It has been said that F=D·x translates into σ=G·ε’=2·G·ε. Check dimensions in both cases (Newtons, Pascals etc.). Compare the dimensions of the right sides of both equations.
2. Consider the Burger model. What is ε(0) in case of constant stress and σ(0) in case of constant strain?
3. Compare single-exponential and stretched-exponential (Kohlrausch) behaviour. If b=0.5, what is the amount of Exp(-t/τ) and Exp(-(t/τ)b) after t=0.1·τ, t=τ, t=10·τ?
4. Which are the three time domains of a constant stress experiment?
5. How is η/G0 related to the Kohlrausch-function Exp(-(t/τx)b)?
MITT Ulrich Fotheringham, SCHOTT AG: Relaxation Processes in Glass and Polymers, Lecture 5
Shear relaxation experiments from Clemson University IV
preliminary results
I. Obtain Strain-time curve through creep recovery experiment
III Characterization of glass in transition region
0 50 100 150 200 250 300 350 4000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1x 10-3
Time(Sec)
Stra
in
0 50 100 150 200 250 300 350 4000
0.2
0.4
0.6
0.8
1
1.2
1.4
Time(Sec)
Disp
lacem
ent(m
m)
Area of interest
(t)*d*k*4+
d*D*k*8= (t) 23 δε
ΠΠ20
Retardation Spectrum on log time scale at different temperatures
10-2
10-1
100
101
102
103
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Time(Sec) on Log Scale
Dis
plac
emen
t(mm
)
a) 522•C
b) 536•C
c) 547•C
22
MITT Ulrich Fotheringham, SCHOTT AG: Relaxation Processes in Glass and Polymers, Lecture 5
Shear relaxation experiments from Clemson University V
preliminary resultsIII. Carryout curve fitting for the resultant retardation curve to determine
retardation parameters i.e. retardation times and retardation weights.
• Retardation function (Prony Series) (m1 = 5 is the best fit)
• Determine Retardation weights •1j (j=1 to 5) & Retardation weights •1j (j=1 to 5)
such that the resultant curve overlaps experimental curve
• A total of 10 parameters are obtained in this process
Retardation Parameters
−∑=Φ= ij
ij
m
j
ttλ
υ exp)(1
1
23
Calculated and Experimental Spectrum (Curve Fitting)
o : Experimental curve
- - - :Simulated curve
10-2 10-1 100 101 102 1030
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Time (sec)
Ret
arda
tion
func
tion
25
MITT Ulrich Fotheringham, SCHOTT AG: Relaxation Processes in Glass and Polymers, Lecture 5
Shear Retardation Parameters
•1j •1j0.0011 0.00150.0124 1.99440.2187 2.48100.2532 8.62570.4939 45.9273
−∑=Φ= ij
ij
m
j
ttλ
υ exp)(1
1
26
Shear relaxation experiments from Clemson University VI
preliminary results
Shear Relaxation Parameters for Pyrex®
w1j •1j0.01133 0.03766383145
0.052928 1.76269908520.148225 2.1680468630.186358 6.9306352740.191795 32.469224890.39312 220.6745301
30